The problem of detecting changes has been receiving considerable attention in various fields. In general, the change-point problem is to identify the location(s) in an ordered sequence that divides this sequence into groups, which follow different models. This dissertation considers the change-point problem in quantile regression for observational or clinical studies involving correlated data (e.g. longitudinal studies) . Our research is motivated by the lack of ideal inference procedures for such models. Our contributions are two-fold. First, we extend the previously reported work on the bent line quantile regression model [Li et al. (2011)] to a longitudinal framework. Second, we propose a score-type test for hypothesis testing of the change-point problem using rank-based inference. The proposed test in this thesis has several advantages over the existing inferential approaches. Most importantly, it circumvents the difficulties of estimating nuisance parameters (e.g. density function of unspecified error) as required for the Wald test in previous works and thus is more reliable in finite sample performance. Furthermore, we demonstrate, through a series of simulations, that the proposed methods also outperform the extensively used bootstrap methods by providing more accurate and computationally efficient confidence intervals. To illustrate the usage of our methods, we apply them to two datasets from real studies: the Finnish Longitudinal Growth Study and an AIDS clinical trial. In each case, the proposed approach sheds light on the response pattern by providing an estimated location of abrupt change along with its 95% confidence interval at any quantile of interest — a key parameter with clinical implications. The proposed methods allow for different change-points at different quantile levels of the outcome. In this way, they offer a more comprehensive picture of the covariate effects on the response variable than is provided by other change-point models targeted exclusively on the conditional mean. We conclude that our framework and proposed methodology are valuable for studying the change-point problem involving longitudinal data.